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<b>let </b><font color="Maroon">c<sub>1</sub></font> be  non <a href="struct_0.html#V3">empty</a> <a href="rlvect_1.html#V4">add-associative</a> <a href="rlvect_1.html#V5">right_zeroed</a> <a href="rlvect_1.html#V6">right_complementable</a> <a href="vectsp_1.html#V4">right-distributive</a>  <a href="vectsp_1.html#L3">doubleLoopStr</a> ;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be  non <a href="struct_0.html#V3">empty</a>  <a href="vectsp_1.html#L4">VectSpStr</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font> be  <a href="bilinear.html#V7" target="_self">symmetric</a> <a href="bilinear.html#NM2">bilinear-Form</a> of <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>2</sub></font>;<br/>





<b>thus </b><a NAME="E1:144"/>
 <a href="bilinear.html#K12" target="_self">leftker</a> <font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a>  <a href="bilinear.html#K13" target="_self">rightker</a> <font color="Maroon">c<sub>3</sub></font>
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="144_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>4</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:144_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="bilinear.html#K12" target="_self">leftker</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>5</sub></font> being   <a href="vectsp_1.html#NM5">Vector</a> of <font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E3:144_1"/><i><font color="Green">E63</font></i>: 
( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="vectsp_1.html#NM5">Vector</a> of <font color="Maroon">c<sub>2</sub></font> holds <font color="Maroon">c<sub>3</sub></font> <a href="binop_1.html#K2">.</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>1</sub></font> ) )
 ;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="144_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>6</sub></font> be   <a href="vectsp_1.html#NM5">Vector</a> of <font color="Maroon">c<sub>2</sub></font>;<br/><b>thus </b><font color="Maroon">c<sub>3</sub></font> <a href="binop_1.html#K2">.</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>5</sub></font> = 
<font color="Maroon">c<sub>3</sub></font> <a href="binop_1.html#K2">.</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font>
<b>by </b><i><a class="ref" href="bilinear.html#D26" target="_self">Def26</a></i>
<br/>.= 
 <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>1</sub></font>
<b>by </b><i><a class="txt" href="bilinear.html#E3:144_1"><i><font color="Green">E63</font></i></a></i>
;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:144_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="bilinear.html#K13" target="_self">rightker</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="bilinear.html#E3:144_1"><i><font color="Green">E63</font></i></a></i>;<br/>


</div><b>end;</b></div>

<b>let </b><font color="Maroon">c<sub>4</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E2:144"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="bilinear.html#K13" target="_self">rightker</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>5</sub></font> being   <a href="vectsp_1.html#NM5">Vector</a> of <font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E4:144"/><i><font color="Green">E63</font></i>: 
( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="vectsp_1.html#NM5">Vector</a> of <font color="Maroon">c<sub>2</sub></font> holds <font color="Maroon">c<sub>3</sub></font> <a href="binop_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>1</sub></font> ) )
 ;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="144_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>6</sub></font> be   <a href="vectsp_1.html#NM5">Vector</a> of <font color="Maroon">c<sub>2</sub></font>;<br/><b>thus </b><font color="Maroon">c<sub>3</sub></font> <a href="binop_1.html#K2">.</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font> = 
<font color="Maroon">c<sub>3</sub></font> <a href="binop_1.html#K2">.</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>5</sub></font>
<b>by </b><i><a class="ref" href="bilinear.html#D26" target="_self">Def26</a></i>
<br/>.= 
 <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>1</sub></font>
<b>by </b><i><a class="txt" href="bilinear.html#E4:144"><i><font color="Green">E63</font></i></a></i>
;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E6:144"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="bilinear.html#K12" target="_self">leftker</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="bilinear.html#E4:144"><i><font color="Green">E63</font></i></a></i>;<br/>


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